midterm2practice
2025-10-24
library(readr)
library(fpp2)## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo## ── Attaching packages ────────────────────────────────────────────── fpp2 2.5 ──## ✔ ggplot2 4.0.0 ✔ fma 2.5
## ✔ forecast 8.24.0 ✔ expsmooth 2.3## TRANSIT <- read_csv("Downloads/TRANSIT.csv")## Rows: 283 Columns: 2## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## dbl (1): TRANSIT
## date (1): DATE
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.Transit_Raw <- TRANSIT$TRANSIT
Transit_ts <- ts(Transit_Raw, frequency = 12, start = c(2000,1))
plot(Transit_ts)
plot(TRANSIT$TRANSIT)
plot(Transit_ts)
#Looking at this time series chart, it is clear that there is seasonality in this transit data. Up until 2020, there seemed to be limited trend. 2020 drastically changed the transits and since then, there has been both seasonality as well as an upward recovery trend.
fivenum(Transit_Raw)## [1] 171450.0 772501.5 824766.0 866163.5 993437.0mean(Transit_Raw)## [1] 779435.2boxplot(Transit_ts)
#Looking at these numbers, it is clear that my data is skewed towards the left tail. Looking at the box plot, it is clear that there are many outliers below the bottom whisker. This is supported by the fact that the median is 824766 but the mean is 779435. These lower outliers are pulling the mean down
decompose(Transit_ts)## $x
## Jan Feb Mar Apr May Jun Jul Aug Sep Oct
## 2000 724934 756536 843730 757458 818246 793911 743746 787064 802516 851255
## 2001 800293 761213 846702 803744 850680 807301 785014 809941 768513 871470
## 2002 794010 756102 820151 824757 838274 772040 782849 781609 805016 877109
## 2003 778679 732135 820157 810703 799984 754298 767411 745756 807125 830771
## 2004 740829 750265 843276 803337 781133 784558 754430 772963 814446 845387
## 2005 756467 750973 842424 816425 806149 795101 753939 808031 858809 858662
## 2006 793138 761980 874469 806944 870445 827752 790273 840294 850328 907084
## 2007 838203 796569 934752 884098 993437 876764 860196 908061 903028 946754
## 2008 852130 835901 889940 908295 912568 884081 910651 890570 929094 971677
## 2009 827360 806287 897860 874039 856958 848878 843229 825429 875511 917949
## 2010 788830 746114 893899 869986 853057 842674 819152 844979 864777 888591
## 2011 793606 780318 907771 856803 880563 861412 824766 843294 891667 919419
## 2012 843287 859851 913814 869911 900348 851686 848905 893636 877531 924987
## 2013 864280 813159 888516 907515 908342 851490 868345 888314 900754 972814
## 2014 832033 812275 913232 920884 923901 874651 886186 887484 931357 986733
## 2015 818017 779983 907646 901828 871727 878666 890344 840937 897493 953141
## 2016 796025 832503 907384 871083 879877 867550 809114 864510 884140 889002
## 2017 802693 777821 886230 834677 875749 842737 790814 837928 844867 910849
## 2018 788672 766031 841125 829115 857964 822283 797080 833069 816289 921343
## 2019 773669 740735 832480 853038 861455 800261 822014 835988 846274 915445
## 2020 815015 788599 505660 171450 200586 256196 306701 318609 331020 351338
## 2021 292833 275803 348362 352403 374081 409969 423379 428647 463998 500798
## 2022 392741 427514 512795 507034 517434 522856 495785 533257 563825 583121
## 2023 525669 512739 593360 560127 609180 572834 532206
## Nov Dec
## 2000 816120 755040
## 2001 820117 756221
## 2002 788746 754257
## 2003 724971 758869
## 2004 797405 767658
## 2005 825130 760996
## 2006 851564 804029
## 2007 870712 808452
## 2008 839026 835899
## 2009 830535 815176
## 2010 837848 812769
## 2011 862730 839221
## 2012 863060 824694
## 2013 862788 837701
## 2014 841871 864282
## 2015 848427 850726
## 2016 839892 807517
## 2017 831770 773762
## 2018 812423 763255
## 2019 814066 791316
## 2020 311158 309585
## 2021 469267 450587
## 2022 549008 520241
## 2023
##
## $seasonal
## Jan Feb Mar Apr May Jun
## 2000 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2001 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2002 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2003 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2004 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2005 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2006 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2007 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2008 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2009 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2010 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2011 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2012 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2013 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2014 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2015 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2016 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2017 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2018 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2019 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2020 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2021 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2022 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2023 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## Jul Aug Sep Oct Nov Dec
## 2000 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2001 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2002 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2003 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2004 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2005 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2006 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2007 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2008 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2009 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2010 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2011 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2012 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2013 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2014 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2015 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2016 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2017 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2018 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2019 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2020 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2021 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2022 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2023 -20385.6561
##
## $trend
## Jan Feb Mar Apr May Jun Jul Aug
## 2000 NA NA NA NA NA NA 790686.3 794021.1
## 2001 803859.0 806531.7 806068.1 805493.6 806502.5 806718.2 806505.6 806030.9
## 2002 801294.0 800023.2 800363.7 802119.6 801047.5 799658.5 798937.9 797300.5
## 2003 789818.6 787681.5 786275.5 784432.6 779844.5 777379.4 775994.5 775172.8
## 2004 777650.9 778243.6 779682.3 780596.3 784223.4 787607.7 788625.5 789306.6
## 2005 793298.5 794739.2 798048.9 800450.5 802158.8 803036.4 804286.8 806273.4
## 2006 818205.2 821063.4 822054.3 823718.5 826837.5 829732.0 833402.7 836721.6
## 2007 866863.0 872600.1 877619.6 881468.3 883919.1 884901.2 885665.8 887884.9
## 2008 883778.8 885152.3 885509.6 887634.1 887352.3 887175.7 887287.2 885021.2
## 2009 871215.7 865692.2 860745.4 856274.1 853681.6 852464.4 849995.5 845882.9
## 2010 840862.5 840673.9 841041.2 839370.8 838452.2 838656.6 838755.3 840379.5
## 2011 845949.7 846113.4 847163.6 849568.5 851889.8 854028.7 857200.9 862584.8
## 2012 869338.6 872442.0 873950.6 873593.6 873839.3 873247.8 873517.2 872446.4
## 2013 872986.2 873574.5 874320.4 877280.8 879262.2 879792.9 878991.2 877610.8
## 2014 884717.7 885426.5 886667.0 888522.1 888230.5 888466.5 888990.1 887060.6
## 2015 879821.6 878055.4 874704.9 871894.2 870767.8 870476.1 868994.9 870266.9
## 2016 866239.6 863837.2 864263.0 861034.2 858006.1 855850.1 854327.6 852327.0
## 2017 842077.7 840207.6 837463.6 836737.5 837309.4 835564.5 833573.9 832498.4
## 2018 824859.4 824918.0 823524.8 822771.3 822402.5 821158.5 820095.6 818416.5
## 2019 818130.3 819290.9 820661.9 821665.5 821488.2 822725.9 825617.8 829334.9
## 2020 625412.7 582383.9 539357.5 494384.1 449925.2 408898.5 367068.8 323944.8
## 2021 336683.4 346129.9 356255.6 368023.8 380839.2 393302.2 403340.1 413824.2
## 2022 471104.4 478480.1 486998.3 494587.9 501340.5 507565.3 516006.2 525096.0
## 2023 553113.0 NA NA NA NA NA NA
## Sep Oct Nov Dec
## 2000 794339.8 796392.2 799672.2 801581.6
## 2001 804711.6 804480.9 804839.5 802853.4
## 2002 796302.1 795716.7 793535.8 791201.1
## 2003 776891.5 777547.9 776455.5 776930.9
## 2004 789300.6 789810.4 791398.1 792879.7
## 2005 808067.2 809007.4 811291.3 815330.8
## 2006 840674.6 846401.2 854740.6 861907.4
## 2007 887656.6 886797.6 884436.3 881371.6
## 2008 884117.3 883020.0 879275.6 875491.7
## 2009 843210.6 842876.7 842545.3 842124.2
## 2010 842382.7 842411.4 843008.2 844935.0
## 2011 866150.5 866948.4 868319.0 868738.1
## 2012 869446.8 869959.6 871859.5 872184.4
## 2013 878603.7 880190.6 881396.0 883009.3
## 2014 885482.3 884455.6 881487.7 879481.0
## 2015 872444.3 871152.4 870210.9 870087.3
## 2016 849167.2 846768.8 845079.9 843874.0
## 2017 830127.8 828016.7 827043.9 825450.6
## 2018 817002.3 817638.9 818781.1 818009.0
## 2019 817711.7 775694.7 719759.0 669553.5
## 2020 296024.2 297009.8 311778.5 325414.6
## 2021 426996.9 440291.2 452707.2 463383.9
## 2022 532003.9 537573.0 543607.9 549513.1
## 2023
##
## $random
## Jan Feb Mar Apr May
## 2000 NA NA NA NA NA
## 2001 23569.96222 438.48833 9253.12848 26.82355 29638.31219
## 2002 19852.00389 1835.94666 -11593.45485 24413.82355 22687.31219
## 2003 15996.37889 -9789.26167 2500.79515 28046.86522 5600.22886
## 2004 -9685.91278 17778.57166 32212.96181 24517.11522 -17629.64614
## 2005 -9695.57944 1990.94666 12994.37848 17750.99022 -10549.02114
## 2006 2068.79556 -13326.17834 21033.96181 -14998.05145 29068.27052
## 2007 -1524.07944 -30273.92834 25751.67015 4406.11522 94978.68719
## 2008 -4512.82944 -3494.09501 -26950.32985 22437.32355 10676.43719
## 2009 -16719.70444 -13648.01167 5733.87848 19541.36522 -11262.85448
## 2010 -24896.57944 -48802.72001 21477.00348 32391.69855 65.56219
## 2011 -25207.70444 -20038.17834 29226.67015 9010.94855 14134.02052
## 2012 1084.33722 33166.19666 8482.67015 -1906.13478 11969.43719
## 2013 18429.71222 -14658.30334 -17185.12152 32010.65689 14540.52052
## 2014 -25548.74611 -27394.30334 -4815.78819 34138.32355 21131.22886
## 2015 -34668.62111 -52315.17834 1560.33681 31710.19855 -13579.97948
## 2016 -43078.62111 14422.98833 11740.21181 11825.24022 7331.64552
## 2017 -12248.70444 -16629.38667 17385.62848 -284.09311 23900.35386
## 2018 -9051.45444 -13129.84501 -13780.57985 8120.11522 21022.31219
## 2019 -17325.37111 -32798.67834 -19562.62152 33148.94855 25427.56219
## 2020 216738.25389 251972.32166 -65078.24652 -321157.67645 -263878.39614
## 2021 -16714.45444 -24569.72001 -39274.32985 -13844.38478 -21297.43781
## 2022 -51227.45444 -5208.88667 -5584.03819 14222.57355 1554.22886
## 2023 -308.07944 NA NA NA NA
## Jun Jul Aug Sep Oct
## 2000 NA -26554.63560 -6767.46531 -14616.70807 -15146.43814
## 2001 11525.15499 -1105.96894 4099.78469 -58991.49973 -3020.06314
## 2002 -16676.13667 4296.78106 -15501.79865 -14078.95807 11383.06186
## 2003 -12139.05334 11802.15606 -29227.17365 7440.58360 -16786.10480
## 2004 7892.65499 -13809.84394 -16153.92365 2352.54193 -14432.60480
## 2005 3006.94666 -29962.13560 1947.28469 27948.91693 -20354.56314
## 2006 8962.40499 -22744.05227 3762.03469 -13139.49973 -9326.35480
## 2007 2805.15499 -5084.13560 20365.74302 -7421.45807 -10052.81314
## 2008 7847.65499 43749.40606 5738.40969 22183.79193 18647.81186
## 2009 7355.98833 13619.15606 -20264.21531 9507.50027 5063.10353
## 2010 14959.73833 782.32273 4789.15969 -398.54140 -23829.56314
## 2011 18325.69666 -12049.21894 -19101.13198 2723.66693 -17538.60480
## 2012 -10619.42834 -4226.55227 21379.24302 -14708.70807 -14981.77147
## 2013 -17360.51167 9739.44773 10892.90969 -642.62473 22614.18686
## 2014 -2873.17834 17581.57273 613.07635 23081.79193 32268.22853
## 2015 19132.27999 41734.73940 -29140.25698 2255.79193 11979.43686
## 2016 22642.23833 -24827.92727 12372.65969 12179.95860 -27776.02147
## 2017 18114.82166 -22374.21894 5619.24302 -8053.66640 12823.14520
## 2018 12066.82166 -2629.96894 14842.15969 -23506.16640 33694.93686
## 2019 -11522.51167 16781.82273 6842.74302 5769.37527 69741.06186
## 2020 -141760.17834 -39982.17727 -5146.09031 12202.95860 -15680.97980
## 2021 27609.19666 40424.57273 15012.45135 14208.25027 -9502.39647
## 2022 26233.02999 164.40606 8350.70135 9028.25027 -24461.14647
## 2023 NA NA
## Nov Dec
## 2000 19052.98759 -16612.06857
## 2001 17882.73759 -16702.86024
## 2002 -2184.51241 -7014.56857
## 2003 -48879.30408 11867.59809
## 2004 8612.15425 4707.80643
## 2005 16443.90425 -24405.27691
## 2006 -571.34575 -27948.90191
## 2007 -11119.05408 -42990.11024
## 2008 -37644.34575 -9663.19357
## 2009 -9405.05408 2981.26476
## 2010 -2554.92908 -2236.48524
## 2011 -2983.72075 412.43143
## 2012 -6194.26241 -17560.90191
## 2013 -16002.72075 -15378.77691
## 2014 -37011.42908 14730.47309
## 2015 -19178.67908 10568.18143
## 2016 -2582.67908 -6427.52691
## 2017 7331.36259 -21759.06857
## 2018 -3752.88741 -24824.48524
## 2019 96912.19592 151692.05643
## 2020 1984.77925 14099.88976
## 2021 19165.02925 17132.63976
## 2022 8005.32092 657.43143
## 2023
##
## $figure
## [1] -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## [7] -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(decompose(Transit_ts))
attributes((decompose(Transit_ts)))## $names
## [1] "x" "seasonal" "trend" "random" "figure" "type"
##
## $class
## [1] "decomposed.ts"stl_transit<-stl(Transit_ts,s.window = "periodic")
plot(stl_transit)
attributes(stl_transit)## $names
## [1] "time.series" "weights" "call" "win" "deg"
## [6] "jump" "inner" "outer"
##
## $class
## [1] "stl"attributes((decompose(Transit_ts)))## $names
## [1] "x" "seasonal" "trend" "random" "figure" "type"
##
## $class
## [1] "decomposed.ts"decomp_transit<-decompose(Transit_ts)
decomp_transit$seasonal## Jan Feb Mar Apr May Jun
## 2000 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2001 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2002 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2003 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2004 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2005 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2006 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2007 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2008 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2009 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2010 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2011 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2012 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2013 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2014 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2015 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2016 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2017 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2018 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2019 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2020 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2021 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2022 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## 2023 -27135.9622 -45757.1967 31380.7465 -1776.4486 14539.2295 -10942.3633
## Jul Aug Sep Oct Nov Dec
## 2000 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2001 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2002 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2003 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2004 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2005 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2006 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2007 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2008 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2009 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2010 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2011 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2012 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2013 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2014 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2015 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2016 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2017 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2018 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2019 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2020 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2021 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2022 -20385.6561 -189.6597 22792.8747 70009.1881 -2605.2376 -29929.5148
## 2023 -20385.6561seasadj(stl_transit)## Jan Feb Mar Apr May Jun Jul Aug
## 2000 753278.5 801529.0 811357.6 760753.0 802351.7 804104.2 765494.7 787285.0
## 2001 828637.5 806206.0 814329.6 807039.0 834785.7 817494.2 806762.7 810162.0
## 2002 822354.5 801095.0 787778.6 828052.0 822379.7 782233.2 804597.7 781830.0
## 2003 807023.5 777128.0 787784.6 813998.0 784089.7 764491.2 789159.7 745977.0
## 2004 769173.5 795258.0 810903.6 806632.0 765238.7 794751.2 776178.7 773184.0
## 2005 784811.5 795966.0 810051.6 819720.0 790254.7 805294.2 775687.7 808252.0
## 2006 821482.5 806973.0 842096.6 810239.0 854550.7 837945.2 812021.7 840515.0
## 2007 866547.5 841562.0 902379.6 887393.0 977542.7 886957.2 881944.7 908282.0
## 2008 880474.5 880894.0 857567.6 911590.0 896673.7 894274.2 932399.7 890791.0
## 2009 855704.5 851280.0 865487.6 877334.0 841063.7 859071.2 864977.7 825650.0
## 2010 817174.5 791107.0 861526.6 873281.0 837162.7 852867.2 840900.7 845200.0
## 2011 821950.5 825311.0 875398.6 860098.0 864668.7 871605.2 846514.7 843515.0
## 2012 871631.5 904844.0 881441.6 873206.0 884453.7 861879.2 870653.7 893857.0
## 2013 892624.5 858152.0 856143.6 910810.0 892447.7 861683.2 890093.7 888535.0
## 2014 860377.5 857268.0 880859.6 924179.0 908006.7 884844.2 907934.7 887705.0
## 2015 846361.5 824976.0 875273.6 905123.0 855832.7 888859.2 912092.7 841158.0
## 2016 824369.5 877496.0 875011.6 874378.0 863982.7 877743.2 830862.7 864731.0
## 2017 831037.5 822814.0 853857.6 837972.0 859854.7 852930.2 812562.7 838149.0
## 2018 817016.5 811024.0 808752.6 832410.0 842069.7 832476.2 818828.7 833290.0
## 2019 802013.5 785728.0 800107.6 856333.0 845560.7 810454.2 843762.7 836209.0
## 2020 843359.5 833592.0 473287.6 174745.0 184691.7 266389.2 328449.7 318830.0
## 2021 321177.5 320796.0 315989.6 355698.0 358186.7 420162.2 445127.7 428868.0
## 2022 421085.5 472507.0 480422.6 510329.0 501539.7 533049.2 517533.7 533478.0
## 2023 554013.5 557732.0 560987.6 563422.0 593285.7 583027.2 553954.7
## Sep Oct Nov Dec
## 2000 779732.2 781205.4 818560.6 784904.2
## 2001 745729.2 801420.4 822557.6 786085.2
## 2002 782232.2 807059.4 791186.6 784121.2
## 2003 784341.2 760721.4 727411.6 788733.2
## 2004 791662.2 775337.4 799845.6 797522.2
## 2005 836025.2 788612.4 827570.6 790860.2
## 2006 827544.2 837034.4 854004.6 833893.2
## 2007 880244.2 876704.4 873152.6 838316.2
## 2008 906310.2 901627.4 841466.6 865763.2
## 2009 852727.2 847899.4 832975.6 845040.2
## 2010 841993.2 818541.4 840288.6 842633.2
## 2011 868883.2 849369.4 865170.6 869085.2
## 2012 854747.2 854937.4 865500.6 854558.2
## 2013 877970.2 902764.4 865228.6 867565.2
## 2014 908573.2 916683.4 844311.6 894146.2
## 2015 874709.2 883091.4 850867.6 880590.2
## 2016 861356.2 818952.4 842332.6 837381.2
## 2017 822083.2 840799.4 834210.6 803626.2
## 2018 793505.2 851293.4 814863.6 793119.2
## 2019 823490.2 845395.4 816506.6 821180.2
## 2020 308236.2 281288.4 313598.6 339449.2
## 2021 441214.2 430748.4 471707.6 480451.2
## 2022 541041.2 513071.4 551448.6 550105.2
## 2023plot(seasadj(stl_transit))
#Looking at the seasonally adjusted data, seasonality plays a rather large role in influencing transit. This data is not only affected by seasonality, but the graph changes drastically after removing seasonality
#Looking at these charts and numbers, it is clear that my data does have seasonality. The decomposition charts show in the seasonality sector that this is present. Looking at the stability of my seasonality chart and the use of stl, I would say that my decomposition is additive. This is also confirmed in the decomp model. The time series is highest in October at around 70000 and lowest in February. Maybe people rely the most on public transportation in October there are not a lot of car sales and in February people got new cars.
naive(Transit_ts)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Aug 2023 532206 460690.8 603721.2 422832.9 641579.1
## Sep 2023 532206 431068.2 633343.8 377529.1 686882.9
## Oct 2023 532206 408338.0 656074.0 342766.3 721645.7
## Nov 2023 532206 389175.6 675236.4 313459.9 750952.1
## Dec 2023 532206 372293.2 692118.8 287640.4 776771.6
## Jan 2024 532206 357030.3 707381.7 264297.8 800114.2
## Feb 2024 532206 342994.6 721417.4 242832.1 821579.9
## Mar 2024 532206 329930.5 734481.5 222852.3 841559.7
## Apr 2024 532206 317660.4 746751.6 204086.8 860325.2
## May 2024 532206 306055.1 758356.9 186338.0 878074.0plot(naive(Transit_ts))
accuracy(naive(Transit_ts))## ME RMSE MAE MPE MAPE MASE
## Training set -683.4326 55803.6 41483.85 -0.6755225 6.063339 0.7298063
## ACF1
## Training set -0.1325041residuals(naive(Transit_ts))## Jan Feb Mar Apr May Jun Jul Aug Sep
## 2000 NA 31602 87194 -86272 60788 -24335 -50165 43318 15452
## 2001 45253 -39080 85489 -42958 46936 -43379 -22287 24927 -41428
## 2002 37789 -37908 64049 4606 13517 -66234 10809 -1240 23407
## 2003 24422 -46544 88022 -9454 -10719 -45686 13113 -21655 61369
## 2004 -18040 9436 93011 -39939 -22204 3425 -30128 18533 41483
## 2005 -11191 -5494 91451 -25999 -10276 -11048 -41162 54092 50778
## 2006 32142 -31158 112489 -67525 63501 -42693 -37479 50021 10034
## 2007 34174 -41634 138183 -50654 109339 -116673 -16568 47865 -5033
## 2008 43678 -16229 54039 18355 4273 -28487 26570 -20081 38524
## 2009 -8539 -21073 91573 -23821 -17081 -8080 -5649 -17800 50082
## 2010 -26346 -42716 147785 -23913 -16929 -10383 -23522 25827 19798
## 2011 -19163 -13288 127453 -50968 23760 -19151 -36646 18528 48373
## 2012 4066 16564 53963 -43903 30437 -48662 -2781 44731 -16105
## 2013 39586 -51121 75357 18999 827 -56852 16855 19969 12440
## 2014 -5668 -19758 100957 7652 3017 -49250 11535 1298 43873
## 2015 -46265 -38034 127663 -5818 -30101 6939 11678 -49407 56556
## 2016 -54701 36478 74881 -36301 8794 -12327 -58436 55396 19630
## 2017 -4824 -24872 108409 -51553 41072 -33012 -51923 47114 6939
## 2018 14910 -22641 75094 -12010 28849 -35681 -25203 35989 -16780
## 2019 10414 -32934 91745 20558 8417 -61194 21753 13974 10286
## 2020 23699 -26416 -282939 -334210 29136 55610 50505 11908 12411
## 2021 -16752 -17030 72559 4041 21678 35888 13410 5268 35351
## 2022 -57846 34773 85281 -5761 10400 5422 -27071 37472 30568
## 2023 5428 -12930 80621 -33233 49053 -36346 -40628
## Oct Nov Dec
## 2000 48739 -35135 -61080
## 2001 102957 -51353 -63896
## 2002 72093 -88363 -34489
## 2003 23646 -105800 33898
## 2004 30941 -47982 -29747
## 2005 -147 -33532 -64134
## 2006 56756 -55520 -47535
## 2007 43726 -76042 -62260
## 2008 42583 -132651 -3127
## 2009 42438 -87414 -15359
## 2010 23814 -50743 -25079
## 2011 27752 -56689 -23509
## 2012 47456 -61927 -38366
## 2013 72060 -110026 -25087
## 2014 55376 -144862 22411
## 2015 55648 -104714 2299
## 2016 4862 -49110 -32375
## 2017 65982 -79079 -58008
## 2018 105054 -108920 -49168
## 2019 69171 -101379 -22750
## 2020 20318 -40180 -1573
## 2021 36800 -31531 -18680
## 2022 19296 -34113 -28767
## 2023plot(residuals(naive(Transit_ts)))
#This plot of residuals shows that the residuals are relatively stable, with the exception of covid. This model does not look good since the residuals look cyclic and large.
hist(residuals(naive(Transit_ts)))
#This plot shows that my residuals are largely negative and concentrated between 1e05 and -1e05. The frequency overall is rather high, meaning this may not be the best model.
naive_fit<-fitted.values(naive(Transit_ts))
naive_resid<-residuals(naive(Transit_ts))
plot(naive_fit,naive_resid)
#This chart shows large clusters in the right of the chart, indicating the model is missing some signals as my residuals increase as my models becomes more fitted.
plot(Transit_ts,naive_resid)
#This plot shows a majority of the plots are located towards the right, meaning that as the actual values increase, there tends to be more residuals. This model does worse as the acutal values increase.
Acf(naive_resid)
#This plot shows that many of the residuals are outside of the blue lines, indicating that this model is auto correlated and missed some important signals. This model did not predict the some of the outlier times well, meaning the model would show more accurate if we eliminated some time periods.
accuracy(naive(Transit_ts))## ME RMSE MAE MPE MAPE MASE
## Training set -683.4326 55803.6 41483.85 -0.6755225 6.063339 0.7298063
## ACF1
## Training set -0.1325041plot(accuracy(naive(Transit_ts)))
naive(Transit_ts,12)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Aug 2023 532206 460690.8 603721.2 422832.9 641579.1
## Sep 2023 532206 431068.2 633343.8 377529.1 686882.9
## Oct 2023 532206 408338.0 656074.0 342766.3 721645.7
## Nov 2023 532206 389175.6 675236.4 313459.9 750952.1
## Dec 2023 532206 372293.2 692118.8 287640.4 776771.6
## Jan 2024 532206 357030.3 707381.7 264297.8 800114.2
## Feb 2024 532206 342994.6 721417.4 242832.1 821579.9
## Mar 2024 532206 329930.5 734481.5 222852.3 841559.7
## Apr 2024 532206 317660.4 746751.6 204086.8 860325.2
## May 2024 532206 306055.1 758356.9 186338.0 878074.0
## Jun 2024 532206 295016.9 769395.1 169456.6 894955.4
## Jul 2024 532206 284470.1 779941.9 153326.6 911085.4plot(naive(Transit_ts,12))
#It hard to draw an real conclusion yet since I do not have a point of comparison, but the mape here is 6.06 for this model. This model predicts the value one year from now will be 532206. The nature of this model produces a flat forecast. This model overal did not do the best job since it does not capture the seasonality present.
plot(Transit_ts)
lines(ma(Transit_ts,3),col="red")
lines(ma(Transit_ts,6),col="blue")
lines(ma(Transit_ts,9),col="green")
#Looking at these charts, the MA3 was the best. This is because it was the most responsive to the volatile changes in transportation. As the ma increased, the model became smoothed out and less responsive.
ses(Transit_ts,12)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Aug 2023 539578.3 468678.2 610478.5 431145.9 648010.7
## Sep 2023 539578.3 447103.9 632052.8 398150.9 681005.8
## Oct 2023 539578.3 429686.5 649470.1 371513.3 707643.4
## Nov 2023 539578.3 414674.8 664481.9 348554.8 730601.9
## Dec 2023 539578.3 401283.0 677873.6 328073.9 751082.8
## Jan 2024 539578.3 389078.2 690078.4 309408.3 769748.4
## Feb 2024 539578.3 377791.5 701365.1 292146.7 787009.9
## Mar 2024 539578.3 367242.4 711914.2 276013.3 803143.4
## Apr 2024 539578.3 357302.8 721853.8 260812.0 818344.6
## May 2024 539578.3 347877.9 731278.7 246397.9 832758.8
## Jun 2024 539578.3 338895.2 740261.5 232659.9 846496.8
## Jul 2024 539578.3 330297.6 748859.1 219511.1 859645.6summary(ses(Transit_ts,12))##
## Forecast method: Simple exponential smoothing
##
## Model Information:
## Simple exponential smoothing
##
## Call:
## ses(y = Transit_ts, h = 12)
##
## Smoothing parameters:
## alpha = 0.8374
##
## Initial states:
## l = 732001.6666
##
## sigma: 55323.68
##
## AIC AICc BIC
## 7782.916 7783.002 7793.852
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -812.0035 55127.84 40313.36 -0.8363806 6.026904 0.7092143
## ACF1
## Training set 0.01607982
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Aug 2023 539578.3 468678.2 610478.5 431145.9 648010.7
## Sep 2023 539578.3 447103.9 632052.8 398150.9 681005.8
## Oct 2023 539578.3 429686.5 649470.1 371513.3 707643.4
## Nov 2023 539578.3 414674.8 664481.9 348554.8 730601.9
## Dec 2023 539578.3 401283.0 677873.6 328073.9 751082.8
## Jan 2024 539578.3 389078.2 690078.4 309408.3 769748.4
## Feb 2024 539578.3 377791.5 701365.1 292146.7 787009.9
## Mar 2024 539578.3 367242.4 711914.2 276013.3 803143.4
## Apr 2024 539578.3 357302.8 721853.8 260812.0 818344.6
## May 2024 539578.3 347877.9 731278.7 246397.9 832758.8
## Jun 2024 539578.3 338895.2 740261.5 232659.9 846496.8
## Jul 2024 539578.3 330297.6 748859.1 219511.1 859645.6#Alpha here is .8374. This represent the weight given to my recent data. The remainder, (1-.8374) is the weight given the historical data. The value of intial state is 732001. The value of sigma is 55323. This value represents the standard deviation, which is relatviely large here since our mean was around 79000.
ses_resid<-residuals(ses(Transit_ts))
ses_resid## Jan Feb Mar Apr May
## 2000 -7067.66664 30452.53069 92146.73071 -71285.46416 49194.29336
## 2001 34614.16470 -33450.43055 80048.69764 -29939.06089 42066.77886
## 2002 26455.00224 -33605.41828 58583.49079 14133.88642 15815.70332
## 2003 16801.87805 -43811.38056 80896.61617 3702.84266 -10116.77805
## 2004 -15183.12841 6966.65034 94144.04025 -24627.62625 -26209.38142
## 2005 -17134.47941 -8280.71297 90104.24357 -11344.64964 -12121.06816
## 2006 20864.68426 -27764.61495 107973.42583 -49964.44626 55374.89548
## 2007 25230.53170 -37530.56340 132079.11406 -29172.95166 104594.37713
## 2008 31730.21001 -11068.47042 52238.84901 26851.00824 8639.98724
## 2009 -12348.05617 -23081.25992 87819.11610 -9538.29761 -18632.28715
## 2010 -30940.70134 -47748.12568 140019.35285 -1140.56918 -17114.49959
## 2011 -24465.23459 -17266.97041 124644.73899 -30696.06160 18767.66198
## 2012 -1102.31972 16384.72121 56627.77400 -34693.18693 24794.57855
## 2013 31905.99626 -45931.88094 67886.74278 30039.94628 5712.62891
## 2014 -12337.27722 -21764.50686 97417.26984 23495.72441 6838.29147
## 2015 -46182.73894 -45545.05620 120255.65508 13740.10772 -27866.33996
## 2016 -56823.32412 27236.38310 79310.66374 -23402.09305 4987.93652
## 2017 -11348.56932 -26717.70564 104063.69279 -34628.31640 35440.12895
## 2018 3677.01227 -22042.97904 71508.97975 -379.94129 28787.20721
## 2019 -19.88805 -32937.23455 86388.16268 34607.97537 14045.56283
## 2020 17623.99339 -23549.67358 -286769.06564 -380849.47215 -32804.49666
## 2021 -17972.10049 -19952.94177 69313.89868 15314.05569 24168.64338
## 2022 -61534.08818 24765.23509 89308.76182 8763.97500 11825.35307
## 2023 -44.74156 -12937.27667 78516.91059 -20463.18727 45724.91351
## Jun Jul Aug Sep Oct
## 2000 -16334.15141 -52821.54944 34727.22268 21099.95694 52170.64927
## 2001 -36537.35424 -28229.35265 20335.84194 -38120.62467 96757.14783
## 2002 -63661.76979 455.19377 -1165.96832 23217.36955 75869.02047
## 2003 -47331.36990 5415.13307 -20774.29501 57990.31565 33077.41376
## 2004 -837.63451 -30264.23098 13610.89393 43696.64500 38047.72350
## 2005 -13019.34311 -43279.43651 47053.13035 58430.61468 9356.02300
## 2006 -33686.95226 -42957.76971 43034.44551 17033.02487 59526.21264
## 2007 -99662.00710 -32776.80340 42534.25484 1884.67499 44032.51928
## 2008 -27081.81202 22165.47535 -16476.05723 35844.37132 48412.64747
## 2009 -11110.31303 -7455.95618 -19012.61985 46989.83052 50080.31974
## 2010 -13166.46350 -25663.36384 21653.16856 23319.62235 27606.65063
## 2011 -16098.66991 -39264.25126 12142.15099 50347.77197 35940.44775
## 2012 -44629.46583 -10039.43537 43098.20894 -9095.60479 45976.71141
## 2013 -55922.91096 7759.82426 21231.04026 15892.96836 74644.79642
## 2014 -48137.83574 3705.97128 1900.73079 44182.13056 62561.68177
## 2015 2406.88150 12069.44976 -47444.05199 48839.80670 63591.19569
## 2016 -11515.77329 -60308.89932 45587.49704 27044.24740 9260.41522
## 2017 -27248.09759 -56354.56896 37948.62038 13110.87778 68114.32350
## 2018 -30999.11374 -30244.62574 31070.08248 -11726.83207 103146.77456
## 2019 -58909.66143 12172.06592 15953.63727 12880.66348 71265.88197
## 2020 50274.75087 58681.57176 21451.83809 15899.87842 22903.92026
## 2021 39818.73349 19886.02874 8502.21873 36733.78162 42774.29915
## 2022 7345.24867 -25876.38537 33263.52322 35977.90417 25147.36495
## 2023 -28909.40350 -45329.75999
## Nov Dec
## 2000 -26650.08361 -65414.30932
## 2001 -35616.63634 -69688.60917
## 2002 -76023.83413 -46853.34441
## 2003 -100420.36425 17565.85911
## 2004 -41794.00432 -36544.28232
## 2005 -32010.35758 -69340.09215
## 2006 -45838.79143 -54990.11735
## 2007 -68880.65067 -73462.59321
## 2008 -124777.27653 -23420.49400
## 2009 -79269.05008 -28251.13900
## 2010 -46253.11679 -32601.50229
## 2011 -50843.72688 -31778.10873
## 2012 -54449.45166 -47221.53565
## 2013 -97885.93916 -41006.94772
## 2014 -134687.10959 505.79316
## 2015 -94371.67178 -13049.39523
## 2016 -47603.90702 -40117.19176
## 2017 -68001.04050 -69067.53541
## 2018 -92144.44191 -64154.16360
## 2019 -89788.47816 -37352.99498
## 2020 -36454.95822 -7501.95194
## 2021 -24574.28474 -22676.70608
## 2022 -30023.08945 -33649.88735
## 2023plot(ses_resid)
#The plot of residuals shows that there may be some seasonality that may not be accounted for due to the cyclic residuals. The model also was not able to account for covid well but was pretty responsive afterwards.
hist(ses_resid)
#This plot indicates that a majority of residuals occur between 1e-05 and 1e05 with the residuals being skewed towards the left tail, meaning there are more negative residuals ie: the model tends to over predict.
fit_ses<-fitted.values(ses(Transit_ts))
ses_resid<-residuals(ses(Transit_ts))
plot(ses_resid)
ses_resid## Jan Feb Mar Apr May
## 2000 -7067.66664 30452.53069 92146.73071 -71285.46416 49194.29336
## 2001 34614.16470 -33450.43055 80048.69764 -29939.06089 42066.77886
## 2002 26455.00224 -33605.41828 58583.49079 14133.88642 15815.70332
## 2003 16801.87805 -43811.38056 80896.61617 3702.84266 -10116.77805
## 2004 -15183.12841 6966.65034 94144.04025 -24627.62625 -26209.38142
## 2005 -17134.47941 -8280.71297 90104.24357 -11344.64964 -12121.06816
## 2006 20864.68426 -27764.61495 107973.42583 -49964.44626 55374.89548
## 2007 25230.53170 -37530.56340 132079.11406 -29172.95166 104594.37713
## 2008 31730.21001 -11068.47042 52238.84901 26851.00824 8639.98724
## 2009 -12348.05617 -23081.25992 87819.11610 -9538.29761 -18632.28715
## 2010 -30940.70134 -47748.12568 140019.35285 -1140.56918 -17114.49959
## 2011 -24465.23459 -17266.97041 124644.73899 -30696.06160 18767.66198
## 2012 -1102.31972 16384.72121 56627.77400 -34693.18693 24794.57855
## 2013 31905.99626 -45931.88094 67886.74278 30039.94628 5712.62891
## 2014 -12337.27722 -21764.50686 97417.26984 23495.72441 6838.29147
## 2015 -46182.73894 -45545.05620 120255.65508 13740.10772 -27866.33996
## 2016 -56823.32412 27236.38310 79310.66374 -23402.09305 4987.93652
## 2017 -11348.56932 -26717.70564 104063.69279 -34628.31640 35440.12895
## 2018 3677.01227 -22042.97904 71508.97975 -379.94129 28787.20721
## 2019 -19.88805 -32937.23455 86388.16268 34607.97537 14045.56283
## 2020 17623.99339 -23549.67358 -286769.06564 -380849.47215 -32804.49666
## 2021 -17972.10049 -19952.94177 69313.89868 15314.05569 24168.64338
## 2022 -61534.08818 24765.23509 89308.76182 8763.97500 11825.35307
## 2023 -44.74156 -12937.27667 78516.91059 -20463.18727 45724.91351
## Jun Jul Aug Sep Oct
## 2000 -16334.15141 -52821.54944 34727.22268 21099.95694 52170.64927
## 2001 -36537.35424 -28229.35265 20335.84194 -38120.62467 96757.14783
## 2002 -63661.76979 455.19377 -1165.96832 23217.36955 75869.02047
## 2003 -47331.36990 5415.13307 -20774.29501 57990.31565 33077.41376
## 2004 -837.63451 -30264.23098 13610.89393 43696.64500 38047.72350
## 2005 -13019.34311 -43279.43651 47053.13035 58430.61468 9356.02300
## 2006 -33686.95226 -42957.76971 43034.44551 17033.02487 59526.21264
## 2007 -99662.00710 -32776.80340 42534.25484 1884.67499 44032.51928
## 2008 -27081.81202 22165.47535 -16476.05723 35844.37132 48412.64747
## 2009 -11110.31303 -7455.95618 -19012.61985 46989.83052 50080.31974
## 2010 -13166.46350 -25663.36384 21653.16856 23319.62235 27606.65063
## 2011 -16098.66991 -39264.25126 12142.15099 50347.77197 35940.44775
## 2012 -44629.46583 -10039.43537 43098.20894 -9095.60479 45976.71141
## 2013 -55922.91096 7759.82426 21231.04026 15892.96836 74644.79642
## 2014 -48137.83574 3705.97128 1900.73079 44182.13056 62561.68177
## 2015 2406.88150 12069.44976 -47444.05199 48839.80670 63591.19569
## 2016 -11515.77329 -60308.89932 45587.49704 27044.24740 9260.41522
## 2017 -27248.09759 -56354.56896 37948.62038 13110.87778 68114.32350
## 2018 -30999.11374 -30244.62574 31070.08248 -11726.83207 103146.77456
## 2019 -58909.66143 12172.06592 15953.63727 12880.66348 71265.88197
## 2020 50274.75087 58681.57176 21451.83809 15899.87842 22903.92026
## 2021 39818.73349 19886.02874 8502.21873 36733.78162 42774.29915
## 2022 7345.24867 -25876.38537 33263.52322 35977.90417 25147.36495
## 2023 -28909.40350 -45329.75999
## Nov Dec
## 2000 -26650.08361 -65414.30932
## 2001 -35616.63634 -69688.60917
## 2002 -76023.83413 -46853.34441
## 2003 -100420.36425 17565.85911
## 2004 -41794.00432 -36544.28232
## 2005 -32010.35758 -69340.09215
## 2006 -45838.79143 -54990.11735
## 2007 -68880.65067 -73462.59321
## 2008 -124777.27653 -23420.49400
## 2009 -79269.05008 -28251.13900
## 2010 -46253.11679 -32601.50229
## 2011 -50843.72688 -31778.10873
## 2012 -54449.45166 -47221.53565
## 2013 -97885.93916 -41006.94772
## 2014 -134687.10959 505.79316
## 2015 -94371.67178 -13049.39523
## 2016 -47603.90702 -40117.19176
## 2017 -68001.04050 -69067.53541
## 2018 -92144.44191 -64154.16360
## 2019 -89788.47816 -37352.99498
## 2020 -36454.95822 -7501.95194
## 2021 -24574.28474 -22676.70608
## 2022 -30023.08945 -33649.88735
## 2023plot(fit_ses,ses_resid)
#In this plot, residuals remain rather constant across all fitted values. There are more residuals located as fitted values increase, but the residuals appear equally large in the lower fitted values. This could mean a few things, like that I have more data points at higher fitted values.
plot(Transit_ts,ses_resid)
#This plot shows my residuals seem equally large regardless of the actual value. This chart alos suggests I have more larger actual values, leading to a cluster, IE: this data has a lot of lower outliers that skews the chert.
acf(ses_resid)
#This plot shows many values outside of the blue lines. This means this model may have missed a pattern or signal or has some serious outliers.
accuracy(ses(Transit_ts))## ME RMSE MAE MPE MAPE MASE
## Training set -812.0035 55127.84 40313.36 -0.8363806 6.026904 0.7092143
## ACF1
## Training set 0.01607982#This model has a mape of 6.02. This is remarkably similar to my first mape with the naive model. This model did not drastically improve from the naive model.
ses(Transit_ts,12)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Aug 2023 539578.3 468678.2 610478.5 431145.9 648010.7
## Sep 2023 539578.3 447103.9 632052.8 398150.9 681005.8
## Oct 2023 539578.3 429686.5 649470.1 371513.3 707643.4
## Nov 2023 539578.3 414674.8 664481.9 348554.8 730601.9
## Dec 2023 539578.3 401283.0 677873.6 328073.9 751082.8
## Jan 2024 539578.3 389078.2 690078.4 309408.3 769748.4
## Feb 2024 539578.3 377791.5 701365.1 292146.7 787009.9
## Mar 2024 539578.3 367242.4 711914.2 276013.3 803143.4
## Apr 2024 539578.3 357302.8 721853.8 260812.0 818344.6
## May 2024 539578.3 347877.9 731278.7 246397.9 832758.8
## Jun 2024 539578.3 338895.2 740261.5 232659.9 846496.8
## Jul 2024 539578.3 330297.6 748859.1 219511.1 859645.6plot(ses(Transit_ts,12))
#Overall, this model is still missing some signals as seen in the residual plot. This model also has a very similar mape to the naive model, not showing any improvements.
sse_holt<-HoltWinters(Transit_ts,beta = NULL,gamma = NULL)
plot(sse_holt)
#My alpha here is .928. This represent the weight given to recent data. My beta is 0. This represents how responsive the model is to trend.Gamma is 1, which represents how responsive the model is to seasonality. My initial level is 551045.439. My initial trend is 1531.450, and my initial seasonality is -9283.195. The initial values represent that values at the start. For example, initial trend shows average differences between actual values in the first season. Initial seasonality shows the deviation from the average in the first season.
resd_sse<-residuals(HoltWinters(Transit_ts,beta = NULL,gamma = NULL))
sd(resd_sse)## [1] 43052.6#sigma is 43052.5
plot(resd_sse)
#This plot shows the best residuals so far. This model did not account for covid well, however the residuals are not as cyclic as the past two models and smaller in values as well.
fitt_sse<-fitted.values(sse_holt)
plot(fitted.values(sse_holt),resd_sse)
acf(resd_sse)
#This graph shows that the model has no trend and has little level until the impact from covid. The seasonality is consistent.I got an error when trying to plot my actual for vs residual values. It says incorrect number of dimensions. My acf plot of residuals shows most of the values are close to the blue lines, with a clear outlier at lag 0. This means that with the exception of lag one, the model is close to including all the signals needed to accurately predict. This iacf residual plot is the best for Holtwinters out of the plots done so far.
forecast_holt<-forecast(sse_holt,h=12)
plot(forecast_holt)
accuracy(forecast_holt)## ME RMSE MAE MPE MAPE MASE
## Training set -2603.178 43051.87 26226.83 -0.8402477 4.26629 0.4613965
## ACF1
## Training set 0.06361874#The mape here is 4.26. This is the most accurate model so far.
forecast_holt<-forecast(sse_holt,h=12)
plot(forecast_holt)
forecast_holt## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Aug 2023 543293.7 488119.6 598467.8 458912.1 627675.2
## Sep 2023 561299.7 486009.9 636589.4 446154.0 676445.4
## Oct 2023 607626.4 516561.0 698691.8 468353.9 746898.9
## Nov 2023 543024.6 438538.8 647510.3 383227.4 702821.7
## Dec 2023 519463.8 403095.3 635832.2 341493.5 697434.0
## Jan 2024 523711.3 396565.8 650856.8 329259.0 718163.6
## Feb 2024 518539.3 381461.4 655617.2 308896.8 728181.8
## Mar 2024 586939.7 440602.1 733277.4 363135.6 810743.8
## Apr 2024 562162.1 407116.7 717207.4 325040.6 799283.5
## May 2024 592045.8 428756.4 755335.2 342316.2 841775.3
## Jun 2024 572859.6 401722.8 743996.3 311128.5 834590.6
## Jul 2024 550583.4 371943.7 729223.1 277377.6 823789.2#Overall, this model was the most accurate using the mape as a measure of accuracy. This model has a mape of 4.2. The predicted value in one year is 550583 in July. This is one of the only models that does not produce a flat forecast due to the use of trend and seasonality.
Accuaracy_Comparison<-list("Naive Mape=6.06","SES Mape=6.02","Holtwinters Mape=4.02")
table(Accuaracy_Comparison)## , , Accuaracy_Comparison.3 = Holtwinters Mape=4.02
##
## Accuaracy_Comparison.2
## Accuaracy_Comparison.1 SES Mape=6.02
## Naive Mape=6.06 1#Looking at my accuracy comparison table, the Holtwinters model has the best Mape. I choose the mape at the accuracy measure of comparison because I find that postive vs negative error does not indciate much since over vs under forecasting is equally bad in this case. I find that mean absolute percent error is holistic as it creates a comparison to the original value. For example, saying you under forecasted by 20 units does not really tell me much on its own. However, if you come to learn that the absolute percent error for this forecast is 80%, suddenly this context highlights the stuggle in my model.
plot(Transit_ts)
#Overall, the transit data shows a large amount of seasonality due to the cyclic patterns. This makes Holtwinters model the best, with the lowest mape of 4.02. In addition, this data also has trend following 2020, which the Holtwinters model is able to account for unlike the naive model. The projected value one year from now using the Holtwinters model is 550583. I think the use of trend after covid would suggest that this number will continue to increase over the next two years until it was fully recouped from covid. R Markdown
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